Scale invariance: Difference between revisions
m (Rearrange strong variant, touch up grammar.) |
(Added in "additive" scale invariance an addition to multiplicative scale invariance.) |
||
Line 1: | Line 1: | ||
Scale invariance can refer to one of two criteria: a cardinal voting method criterion and an ordinal one. |
Scale invariance can refer to one of two criteria: a cardinal voting method criterion and an ordinal one, and these can be multiplicative or additive. |
||
The [[Cardinal voting systems#Scale%20invariance|cardinal method criterion]] |
The [[Cardinal voting systems#Scale%20invariance|cardinal method criterion]] (multiplicative version): |
||
{{Definition|Multiplying every ballot's score of every candidate by a constant <math>\alpha>0</math> should not change the outcome.}} |
{{Definition|Multiplying every ballot's score of every candidate by a constant <math>\alpha>0</math> should not change the outcome.}}The additive version: |
||
{{Definition|Adding a constant <math>\alpha</math> to every ballot's score of every candidate should not change the outcome.}} |
|||
⚫ | |||
{{Definition|Multiplying one or more ballot's score of every candidate by a constant <math>\alpha>0</math> should not change the outcome.}}Additive: |
|||
⚫ | |||
{{Definition|Adding a constant <math>\alpha</math> to one or more ballot's score of every candidate should not change the outcome.}} |
|||
⚫ | |||
{{Definition| |
{{Definition|For every way of ranking the candidates, multiplying the number of voters who express this preference by a constant <math>\alpha>0</math> should not change the outcome.}} |
||
Additive: |
|||
⚫ | |||
{{Definition|For every way of ranking the candidates, |
{{Definition|For every way of ranking the candidates, adding to the number of voters who express this preference by a constant <math>\alpha>0</math> should not change the outcome.}} |
||
These criteria represent a desideratum that the method should not rely on absolute numbers when selecting a winner, just on the candidates' or factions' relative support. |
These criteria represent a desideratum that the method should not rely on absolute numbers when selecting a winner, just on the candidates' or factions' relative support. |
Latest revision as of 21:42, 9 October 2021
Scale invariance can refer to one of two criteria: a cardinal voting method criterion and an ordinal one, and these can be multiplicative or additive.
The cardinal method criterion (multiplicative version):
Multiplying every ballot's score of every candidate by a constant should not change the outcome.
The additive version:
Adding a constant to every ballot's score of every candidate should not change the outcome.
with a stronger variant being (multiplicative):
Multiplying one or more ballot's score of every candidate by a constant should not change the outcome.
Additive:
Adding a constant to one or more ballot's score of every candidate should not change the outcome.
The ordinal method criterion is also called the homogeneity criterion. It is (multiplicative):
For every way of ranking the candidates, multiplying the number of voters who express this preference by a constant should not change the outcome.
Additive:
For every way of ranking the candidates, adding to the number of voters who express this preference by a constant should not change the outcome.
These criteria represent a desideratum that the method should not rely on absolute numbers when selecting a winner, just on the candidates' or factions' relative support.
This page is a stub - please add to it.